CHAPTER 1: SINUSOIDS AND PHASORS Sinusoids, Phasors and Phasors Relationships for Circuit Element Ohms & Kirchhoff’s Law in the Frequency Domain. Impedance of Series & Parallel Circuit and delta-wye transformation

BEE1123[CH1]: Sinusoids and Phasor Sinusoidal A sinusoidal voltage source supplies a voltage that varies with time A general expression for a sinusoidal voltage is rms = root mean square NHA BEE1123[CH1]: Sinusoids and Phasor

BEE1123[CH1]: Sinusoids and Phasor Sinusoidal Vm sin ωt: (a) as a function of ωt, (b) as a function of t ω = angular frequency (rad/s) f = the number of cycle per second (Hz) T = the time taken to complete one cycle (s) Vm = the maximum value for the voltage or amplitude of the sinusoidal voltage. = the phase angle of the sinusoidal function (degree or radians) NHA BEE1123[CH1]: Sinusoids and Phasor

BEE1123[CH1]: Sinusoids and Phasor Sinusoidal π = 180o NHA BEE1123[CH1]: Sinusoids and Phasor

BEE1123[CH1]: Sinusoids and Phasor Sinusoidal Let examine the two sinusoids v2 leads v1 by or v1 lags v2 NHA BEE1123[CH1]: Sinusoids and Phasor

BEE1123[CH1]: Sinusoids and Phasor Sinusoidal Example 1 For the following sinusoidal voltage , find the value v at time t= 0s and t=0.5s V = 6 cos(100t + 60o) Solution At t = 0s, At t = 0.5s V = 6 cos( 0 + 60o) V = 6 cos(50 radian + 60o) = 3 V = 4.26 V Be careful when adding ωt and phase angel. The unit ωt is radians. The unit for is degrees. To add both the values, they should be the same units. NHA BEE1123[CH1]: Sinusoids and Phasor

BEE1123[CH1]: Sinusoids and Phasor Sinusoidal Radians Degree π = 3.142 NHA BEE1123[CH1]: Sinusoids and Phasor

BEE1123[CH1]: Sinusoids and Phasor EXAMPLE 1 A sinusoidal current has a maximum amplitude of 20A. The current passes through one complete cycle in 1 ms. The magnitude of the current at zero time is 10A. a) What is the frequency of the current in Hz? b) What is the value of the angular frequency? c) Write the expression for i(t) in the form Im cos(ωt + ). d) What is the rms value of the current? NHA BEE1123[CH1]: Sinusoids and Phasor

BEE1123[CH1]: Sinusoids and Phasor EXAMPLE 2 A sinusoidal voltage is given by the expression v = 300cos(120t + 30o) a) What is the frequency in Hz? b) What is the period of the voltage in milliseconds? c) What is the magnitude of v at t = 2.778 ms? d) What is the rms value of v? NHA BEE1123[CH1]: Sinusoids and Phasor

BEE1123[CH1]: Sinusoids and Phasor Phasors A phasor is a complex number that represents the magnitude and phase of a sinusoidal voltage or current. COMPLEX NUMBER POLAR EXPONENTIAL RECTANGULAR NHA BEE1123[CH1]: Sinusoids and Phasor

BEE1123[CH1]: Sinusoids and Phasor Phasors Example: COMPLEX NUMBER POLAR RECTANGULAR * TRY switch from the polar to rectangular form and vice-versa using the calculator NHA BEE1123[CH1]: Sinusoids and Phasor

Complex Numbers Polar: z   = A = x + jy : Rectangular imaginary axis real axis imaginary axis x is the real part y is the imaginary part z is the magnitude  is the phase NHA BEE1123[CH1]: Sinusoids and Phasor

Complex Number Addition and Subtraction Addition is most easily performed in rectangular coordinates: A = x + jy B = z + jw A + B = (x + z) + j(y + w) Subtraction is also most easily performed in rectangular coordinates: A - B = (x - z) + j(y - w) NHA BEE1123[CH1]: Sinusoids and Phasor

Complex Number Multiplication and Division Multiplication is most easily performed in polar coordinates: A = AM  q B = BM  f A  B = (AM  BM)  (q + f) Division is also most easily performed in polar coordinates: A / B = (AM / BM)  (q - f) NHA BEE1123[CH1]: Sinusoids and Phasor

BEE1123[CH1]: Sinusoids and Phasor Complex Number If the sinusoidal voltage represented by the sine function then subtract the phase, to 90o. As example: Given v(t) = Vm sin (ωt +10o). Transform to phasor. v(t) = Vm sin (ωt +10o) v(t) = Vm cos (ωt + 10o - 90o) v(t) = Vm cos (ωt – 80o) NHA BEE1123[CH1]: Sinusoids and Phasor

BEE1123[CH1]: Sinusoids and Phasor EXAMPLE 3 Given y1= 20cos(100t - 30°) and y2 = 40cos(l00t + 60°). Express y1 + y2 as a single cosine function. NHA BEE1123[CH1]: Sinusoids and Phasor

BEE1123[CH1]: Sinusoids and Phasor Impedance AC steady-state analysis using phasors allows us to express the relationship between current and voltage using a formula that looks likes Ohm’s law: V = I Z Z is called impedance (units of ohms, Ω) Impedance is (often) a complex number, but is not a phasor Impedance depends on frequency, ω NHA BEE1123[CH1]: Sinusoids and Phasor

Phasor Relationships for Circuit Elements Resistor In the phasor domain, Z = R. Voltage-current relationship is given by V = IR . NHA BEE1123[CH1]: Sinusoids and Phasor

Phasor Relationships for Circuit Elements Inductor In the phasor domain, Z = jωL Voltage-current relationship is given by V = jωLI NHA BEE1123[CH1]: Sinusoids and Phasor

Phasor Relationships for Circuit Elements Capacitor In the phasor domain, Voltage-current relationship is give by NHA BEE1123[CH1]: Sinusoids and Phasor

BEE1123[CH1]: Sinusoids and Phasor Impedance A general expression for impedance , Z Z = R ± jX Ω R = resistance , X = reactance NHA BEE1123[CH1]: Sinusoids and Phasor

BEE1123[CH1]: Sinusoids and Phasor Impedance Summary Element Impedance Resistance Reactance Resistor ZR = R = R  0 R Inductor ZL = jL = L  90 L Capacitor ZC = 1 / jC = 1/C  -90 1 / jC NHA BEE1123[CH1]: Sinusoids and Phasor

BEE1123[CH1]: Sinusoids and Phasor EXAMPLE 5 The current in the 75mH inductor is 4cos(40000t – 38o)mA. Calculate a) the inductive reactance b) the impedance of the inductor c) the phasor voltage V d) the steady-state expression v(t). NHA BEE1123[CH1]: Sinusoids and Phasor

BEE1123[CH1]: Sinusoids and Phasor EXAMPLE Find the phasor domain equivalent circuit for the following circuit NHA BEE1123[CH1]: Sinusoids and Phasor

BEE1123[CH1]: Sinusoids and Phasor Series Impedance Z1 Z2 Zeq Z3 Zeq = Z1 + Z2 + Z3 NHA BEE1123[CH1]: Sinusoids and Phasor

BEE1123[CH1]: Sinusoids and Phasor Parallel Impedance Z1 Z2 Z3 Zeq 1/Zeq = 1/Z1 + 1/Z2 + 1/Z3 NHA BEE1123[CH1]: Sinusoids and Phasor

BEE1123[CH1]: Sinusoids and Phasor Exercise Find the input impedance of the circuit. Assume that the circuit operates at ω = 50 rad/s. NHA BEE1123[CH1]: Sinusoids and Phasor

Circuit Analysis Techniques While Obeying Passive Sign Convention Ohm’s Law; KCL; KVL Voltage and Current Division Series/Parallel Impedance combinations NHA BEE1123[CH1]: Sinusoids and Phasor

Kirchoff’s Current Law (KCL) i1(t) i5(t) i2(t) i4(t) i3(t) The sum of currents entering the node is zero: Analogy: mass flow at pipe junction NHA BEE1123[CH1]: Sinusoids and Phasor

Kirchoff’s Voltage Law (KVL) + - v(t)1 v(t)2 v(t)3 The sum of voltages around a loop is zero: Analogy: pressure drop thru pipe loop NHA BEE1123[CH1]: Sinusoids and Phasor

BEE1123[CH1]: Sinusoids and Phasor EXAMPLE 9 A 90Ω, a 32mH inductor and a 5µF capacitor are connected in series across the terminal of a sinusoidal voltage, a shown in Fig 9(a) below. The steady-state expression for the source Vs is 750cos(5000t + 30o)V. a) Construct the phasor domain equivalent circuit b) Calculated the steady-state current i(t) by the phasor method NHA BEE1123[CH1]: Sinusoids and Phasor

BEE1123[CH1]: Sinusoids and Phasor VOLTAGE DIVIDER NHA BEE1123[CH1]: Sinusoids and Phasor

BEE1123[CH1]: Sinusoids and Phasor EXAMPLE 10 Use the concept of voltage division to find the steady-state expression for vo(t) in the circuit if the vg = 100cos(8000t) V NHA BEE1123[CH1]: Sinusoids and Phasor

BEE1123[CH1]: Sinusoids and Phasor CURRENT DIVIDER NHA BEE1123[CH1]: Sinusoids and Phasor

BEE1123[CH1]: Sinusoids and Phasor Exercise Determine vo(t) in the circuit using current divider NHA BEE1123[CH1]: Sinusoids and Phasor

BEE1123[CH1]: Sinusoids and Phasor Delta(Δ)-Wye(Y) a Z1 Zb Zc Z3 Z2 c b Za NHA BEE1123[CH1]: Sinusoids and Phasor

BEE1123[CH1]: Sinusoids and Phasor Wye(Y)-Delta(Δ) a b c Za Zc Zb Z1 Z2 Z3 NHA BEE1123[CH1]: Sinusoids and Phasor

BEE1123[CH1]: Sinusoids and Phasor Exercise Find current I in the circuit. NHA BEE1123[CH1]: Sinusoids and Phasor

BEE1123[CH1]: Sinusoids and Phasor Exercise NHA BEE1123[CH1]: Sinusoids and Phasor